Following closely the steps taken in [1]}, [2]},
the well-known decomposition relation \(R=^{(3)}R+K_{ab}K^{ab}-K^{2}+2\nabla _{b}(Kn^{b}-a^{b})\) 
for the Ricci scalar can be used to re-write the given modified Einstein-Hilbert
action functional in the form
\(S_{G}[g] & =\frac{1}{16\pi }(\underset{M}{\int }[^{(3)}R+K_{ab}K^{ab}-K^{2}]\omega _{g}+\\+ & 2\underset{\mathcal {B}}{\int }[\Theta +\eta K-a_{b}u^{b}]\omega _{\gamma }+\underset{\Omega _{1}}{2\overset{\Omega _{2}}{\int }}\sinh ^{-1}\eta \omega _{q}).\nonumber \) 
